44 research outputs found
Minimization of convex functionals over frame operators
We present results about minimization of convex functionals defined over a finite set of vectors in a finite-dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original oneFil: Massey, Pedro Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; ArgentinaFil: Ruiz, Mariano Andres. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentin
Minimization of convex functionals over frame operators
We present results about minimization of convex functionals defined over a finite set of vectors in a finite-dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one.Facultad de Ciencias Exacta
Optimal reconstruction systems for erasures and for the q-potential
We introduce the -potential as an extension of the Benedetto-Fickus frame
potential, defined on general reconstruction systems and we show that protocols
are the minimizers of this potential under certain restrictions. We extend
recent results of B.G. Bodmann on the structure of optimal protocols with
respect to 1 and 2 lost packets where the worst (normalized) reconstruction
error is computed with respect to a compatible unitarily invariant norm. We
finally describe necessary and sufficient (spectral) conditions, that we call
-fundamental inequalities, for the existence of protocols with prescribed
properties by relating this problem to Klyachko's and Fulton's theory on sums
of hermitian operators
Robust dual reconstruction systems and fusion frames
We study the duality of reconstruction systems, which are g-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.Fil: Massey, Pedro Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Ruiz, Mariano Andres. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Aliasing and oblique dual pair designs for consistent sampling
In this paper we study some aspects of oblique duality between finite
sequences of vectors \cF and \cG lying in finite dimensional subspaces
\cW and \cV, respectively. We compute the possible eigenvalue lists of the
frame operators of oblique duals to \cF lying in \cV; we then compute the
spectral and geometrical structure of minimizers of convex potentials among
oblique duals for \cF under some restrictions. We obtain a complete
quantitative analysis of the impact that the relative geometry between the
subspaces \cV and \cW has in oblique duality. We apply this analysis to
compute those rigid rotations for \cW such that the canonical oblique
dual of U\cdot \cF minimize every convex potential; we also introduce a
notion of aliasing for oblique dual pairs and compute those rigid rotations
for \cW such that the canonical oblique dual pair associated to U\cdot \cF
minimize the aliasing. We point out that these two last problems are intrinsic
to the theory of oblique duality.Comment: 23 page
Duality in reconstruction systems
We consider reconstruction systems (RS's), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS's. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS's, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS's the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.Fil: Massey, Pedro Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Ruiz, Mariano Andres. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin